### Abstract

The envelope of narrow-banded, periodic, surface-gravity waves propagating in one dimension over water of finite, non-uniform depth may be modeled by the Djordjević and Redekopp ["On the development of packets of surface gravity waves moving over an uneven bottom," Z. Angew. Math. Phys. 29, 950-962 (1978)] equation (DRE). Here we find five approximate solutions of the DRE that are in the form of Jacobi-elliptic functions and discuss them within the framework of ocean swell. We find that in all cases, the maximum envelope-amplitude decreases/increases when the wave group propagates on water of decreasing/increasing depth. In the limit of the elliptic modulus approaching one, three of the solutions reduce to the envelope soliton solution. In the limit of the elliptic modulus approaching zero, two of the solutions reduce to an envelope-amplitude that is uniform in an appropriate reference frame.

Original language | English (US) |
---|---|

Article number | 042106 |

Journal | Physics of Fluids |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Apr 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Computational Mechanics
- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Fluid Flow and Transfer Processes

### Cite this

*Physics of Fluids*,

*28*(4), [042106]. https://doi.org/10.1063/1.4945048

}

*Physics of Fluids*, vol. 28, no. 4, 042106. https://doi.org/10.1063/1.4945048

**Periodic envelopes of waves over non-uniform depth.** / Rajan, Girish K.; Bayram, Saziye; Henderson, Diane Marie.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Periodic envelopes of waves over non-uniform depth

AU - Rajan, Girish K.

AU - Bayram, Saziye

AU - Henderson, Diane Marie

PY - 2016/4/1

Y1 - 2016/4/1

N2 - The envelope of narrow-banded, periodic, surface-gravity waves propagating in one dimension over water of finite, non-uniform depth may be modeled by the Djordjević and Redekopp ["On the development of packets of surface gravity waves moving over an uneven bottom," Z. Angew. Math. Phys. 29, 950-962 (1978)] equation (DRE). Here we find five approximate solutions of the DRE that are in the form of Jacobi-elliptic functions and discuss them within the framework of ocean swell. We find that in all cases, the maximum envelope-amplitude decreases/increases when the wave group propagates on water of decreasing/increasing depth. In the limit of the elliptic modulus approaching one, three of the solutions reduce to the envelope soliton solution. In the limit of the elliptic modulus approaching zero, two of the solutions reduce to an envelope-amplitude that is uniform in an appropriate reference frame.

AB - The envelope of narrow-banded, periodic, surface-gravity waves propagating in one dimension over water of finite, non-uniform depth may be modeled by the Djordjević and Redekopp ["On the development of packets of surface gravity waves moving over an uneven bottom," Z. Angew. Math. Phys. 29, 950-962 (1978)] equation (DRE). Here we find five approximate solutions of the DRE that are in the form of Jacobi-elliptic functions and discuss them within the framework of ocean swell. We find that in all cases, the maximum envelope-amplitude decreases/increases when the wave group propagates on water of decreasing/increasing depth. In the limit of the elliptic modulus approaching one, three of the solutions reduce to the envelope soliton solution. In the limit of the elliptic modulus approaching zero, two of the solutions reduce to an envelope-amplitude that is uniform in an appropriate reference frame.

UR - http://www.scopus.com/inward/record.url?scp=84964978167&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964978167&partnerID=8YFLogxK

U2 - 10.1063/1.4945048

DO - 10.1063/1.4945048

M3 - Article

AN - SCOPUS:84964978167

VL - 28

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 4

M1 - 042106

ER -